One consequence could be that Naruto has some new powers or abilities. Since the Shinigami is a very powerful and mysterious being, its intervention might leave a mark on Naruto in the form of newfound powers.
Perhaps Naruto's relationship with the other characters changes. Some might be wary of him now that he has a connection with the Juubi. On the other hand, some might see him as an even greater hero for surviving with the help of such a powerful entity. Also, Naruto might have a new sense of purpose. He could feel obligated to use the newfound power or knowledge from the Juubi to protect the village in a different way, or to seek a greater understanding of the Juubi's origin and the secrets it holds.
Well, in some fanfictions, the Shinigami might save Naruto by using its special powers to reverse a fatal curse or injury. For example, it could extract a harmful entity that was about to kill Naruto from his body.
Maybe the Shinigami uses its special powers to heal Naruto's severe injuries. For example, it could transfer some of its own spiritual energy to Naruto to revive his fading life force.
Maybe it's a new villain created specifically for the fanfiction. This villain could have powers that are extremely dangerous to Naruto, like the ability to drain his chakra completely. So the Shinigami has to step in to save Naruto from this new and powerful foe.
Naruto may be forced to fight in battles or do tasks that he doesn't want to do. His skills and powers could be misused by Rias for her own purposes. Also, his relationships with his friends in his own world might be affected as he is enslaved in another world. He may not be able to return to his home or communicate with his friends, which could cause a lot of emotional distress for him.
Naruto might become more determined. He's the type who turns negative things into positive motivation. So, Sakura's meanness could make him strive harder to prove himself.
One consequence could be that he might face more discrimination from the village elders. In the original story, as Naruto advanced in rank, he gained more respect. But as a genin here, he could still be seen as a troublemaker. Another consequence is that his training might be different. He might focus more on basic genin - level skills, which could make him even stronger in the long run as he masters them to a higher degree.
He might start to compare the fanfiction world too much with the real Naruto story, which could lead to some confusion. For example, if a fanfiction has a completely different view of a character's personality, he may struggle to separate it from the original.
She might spend a lot of time reading and writing fanfiction, which could lead to less time for other activities like socializing or doing school/work - related tasks.
He might have physical injuries that could slow him down in his training or his future battles. For instance, if he gets a deep wound on his leg, it would affect his mobility.